To understand what people refer to in regard to Greek drama, you can read Aristotle's Poetics. It's a great read and very smooth, funny, and soooo interesting. Read that work and not commentaries. Commentaries obfuscate anything he says but he says what he says SO CLEARLY there is no need for commentary until you finish and maybe just look up a few historical terms.
Now, as to before Aristotle, you would have to read the Greek tragedies. They are STUPENDOUS. Absolutely riveting. Euripides will have your hair standing on end.
Now AFTER Aristotle, there is a crap load out there dating centuries back and everyone started to use Aristotle as the grand description of Drama. In fact, Shakespeare's plays are as close to Aristotle's requirements as one could get. For instance, the tragic figure must be wealthy and elite (Hamlet). There must be a reversal (Oedipus realizes his wife is him mom and his kids are now his siblings), a recognition (Odysseus' scar) and then loads of mayhem. Babies are thrown over walls (The Trojan Women), people are enslaved who were queens (Andromache, wife of Hector), people are slashed to pieces (too many to list) kill themselves (Aias), etc etc etc.......it's a blood bath.
Now comedy is awesome, too. Read some Aristophanes if you want to make yourself laugh. He wrote DURING the Peloponnesian War and was advocating PEACE! One guys even makes his own peace treaty (The Acharnians) and some women decide to make the rule since the men only want to fight (The Assembly Women). And of course, there is Lysistrata, perhaps his most well know. He did get sued twice by Cleon, so their freedom of speech had limits. But he bounced back.
Anyway, those Athenians left us a lot to think about and have fun with. Hope this helps!!